2. Concepts and Objectives
Synthetic Division
Review performing synthetic division
Evaluate polynomial functions using the remainder
theorem
3. Dividing Polynomials
Back in section 0.3, we found the quotient of two
polynomials by using something like long division.
More formally, we can state that:
Let fx and gx be polynomials with gx of degree one
or more, but of lower degree than fx. There exist
unique polynomials qx and rx such that
where either rx = 0 or the degree of rx is less than the
degree of gx.
f x g x q x r x
4. Dividing Polynomials (cont.)
For example, could be evaluated as
or
3 2
2
3 2 150
4
x x
x
2 3 2
3
4 3 2 0 150
x
x x x x
3 2
3 0 12x x x
2
2 12 150x x
2
2
2 0 8x x
12 158x
2
12 158
3 2
4
x
x
x
5. Dividing Polynomials (cont.)
Using the division algorithm, this means that
3 2 2
3 2 150 4 3 2 12 158x x x x x
f x g x q x r x
Dividend = Divisor • Quotient + Remainder
6. Synthetic Division
A shortcut method of performing long division with
certain polynomials, called synthetic division, is used
only when a polynomial is divided by a binomial of the
form x – k, where the coefficient of x is 1.
To use synthetic division:
The numbers on the bottom are the coefficients of the
quotient.
1 1 0...n nk a a a a
1
1 1 0...n n
n na x a x a x a
x k
an
kan
1n na ka …
8. Synthetic Division (cont.)
Example: Use synthetic division to divide
3 2
4 15 11 10
3
x x x
x
3 4 15 11 10
9. Synthetic Division (cont.)
Example: Use synthetic division to divide
3 2
4 15 11 10
3
x x x
x
3 4 15 11 10
4
12
–3
10. Synthetic Division (cont.)
Example: Use synthetic division to divide
3 2
4 15 11 10
3
x x x
x
3 4 15 11 10
4
12
–3
–9
2
11. Synthetic Division (cont.)
Example: Use synthetic division to divide
3 2
4 15 11 10
3
x x x
x
3 4 15 11 10
4
12
–3
–9
2
6
–4
12. Synthetic Division (cont.)
Example: Use synthetic division to divide
3 2
4 15 11 10
3
x x x
x
3 4 15 11 10
4
12
–3
–9
2
6
–4
2 4
4 3 2
3
x x
x
Notice that the
exponent is 1 less
than the numerator.
14. Synthetic Division (cont.)
Example: Use synthetic division to divide
4 3
3 15 50 25
4
x x x
x
4 3 15 500 25
Notice the place-
holder for the
missing x2 term.
15. Synthetic Division (cont.)
Example: Use synthetic division to divide
4 3
3 15 50 25
4
x x x
x
4 3 15 500 25
3
–12
3
12
12
48
2
–8
17
3 2 17
3 3 12 2
4
x x x
x
Notice the place-
holder for the
missing x2 term.
17. Synthetic Division (cont.)
Example: Use synthetic division to divide
4 3 2
5 4 3 9
3
x x x x
x
3 1 5 4 3 9
1
–3
2
–6
–2
6
3
–9
0
3 2
2 2 3x x x
18. Remainder Theorem
The remainder theorem:
Example: Let . Find f–3.
If the polynomial fx is divided by x – k, then the
remainder is equal to fk.
4 2
3 4 5f x x x x
19. Remainder Theorem
The remainder theorem:
Example: Let . Find f–3.
If the polynomial fx is divided by x – k, then the
remainder is equal to fk.
4 2
3 4 5f x x x x
3 1 0 3 4 5
–1
3
3
–9
–6
18
14
–42
–47
3 47f
20. Potential Zeros
A zero of a polynomial function f is a number k such that
fk = 0. The real number zeros are the x-intercepts of
the graph of the function.
The remainder theorem gives us a quick way to decide if
a number k is a zero of a polynomial function defined by
fx. Use synthetic division to find fk ; if the remainder
is 0, then fk = 0 and k is a zero of fx .
21. Potential Zeros (cont.)
Example: Decide whether the given number k is a zero
of fx:
4 3 2
4 14 36 45; 3f x x x x x k
22. Potential Zeros (cont.)
Example: Decide whether the given number k is a zero
of fx:
Since the remainder is zero, –3 is a zero of the function.
4 3 2
4 14 36 45; 3f x x x x x k
3 1 4 14 36 45
1
–3
–7
21
7
–21
15
–45
0